From my intro thread (and some stuff on Mud) it seems there's some interest in how carbs work. Since I'm going to be tuning mine soon and don't have a lot of experience with progressive carburetors, I figured I'd learn how these particular carbs are wired by teaching how they work. Bear in mind, I'm new to this particular carb, so don't take this for gospel. If I'm corrected on something, I'll edit to reflect.

The carburetor here is from my rock-stock 11/73 40-Series (this particular carb was actually built on my birthday a few years removed). It's recently rebuilt, but I didn't go so far as to soda blast it or anything, and between a crappy plunger boot and a dirty fuel tank it's already accumulated some schmutz (though some of the spots are just spots on the metal).

Essentials
Carburetors work via pressure differential. Air moving over an orifice (the inside of which is atmospheric pressure) creates a negative pressure zone. That negative pressure draws air/fluid toward the stream (think of those old-timey perfume bottles with the squeeze bulb). In a carburetor, this is taken advantage of via the bowl.
Fuel in the bowl is at atmospheric pressure (or slightly higher) courtesy of passages like those pictured above (from the air filter at the top of the air horn through the brass tubes in the first photo into the bowl via the hole under the float and behind the power valve plunger in the second photo), and there are various channels with apertures leading to the carb throat through which air is flowing at speed by means of the vacuum created by the retracting pistons in their cylinders.
Venturis (the primary venturis are pictured above) and throttle plates are used in the carb throat to control, manipulate, and optimize the air speed (and thus the pressure differentials) in the carburetor over these various apertures at various engine speeds.
Meanwhile, the airflow to a given cylinder effectively has a very limited and finite time to be properly charged with fuel. That time is dictated by the cam and defined by the amount of time the intake valve is open. During that time, the carburetor controls how much air is allowed in (it's always a bit restrictive in order to have sufficient rate of air flow to create enough vacuum to draw fuel from the bowl), and the number, arrangement, size, etc. of the holes in the carb throat dictate pressure differentials at that occur between the bowl and the throat.

LinkageChoke
When the choke cable is pulled (1), it turns the choke linkage (2) which is on a shaft with the choke throttle plate. The choke closes to build extra vacuum in the carburetor to draw more fuel from the float. Articulation of the accelerator pedal (4) or a vacuum at the vacuum pot (also 4) can actuate the choke linkage to open the choke a bit and let more air flow into the engine. This opening occurs at a rate that continues the same percentage increase in fuel for operation required for warm up. As more air is required by the cylinders, it creates a vacuum commensurate with the amount of vacuum lost due to the opening of the choke throttle plate.
The adjustment screw (pictured above) is for fine tuning the choke plate's opening as it relates to the throttle linkage. Gross adjustments are made by utilizing the second hole for the associated rod or bending the linkage.

Choke Breaker
When properly adjusted, the choke is fully closed by the linkage when it is operated by pulling the knob out all the way inside the cab. However, once the engine fires, there wouldn't be enough airflow to keep the engine running. The choke breaker vacuum diaphragm is connected directly to manifold vacuum below the throttle plates. As soon as the engine fires, manifold vacuum is applied to the vacuum diaphragm and it opens the choke plate just enough to allow enough air in to run the engine at the high idle setting (set by the high idle adjustment screw).
The linkage circled above is actuated by the vacuum pot.
The idle set screw (1) rests on the metal stop. The stop is articulated by the vacuum actuated linkage above.

EDIT: Changed to reflect Jeff Zepp's correction. Thanks!

Primary to Secondary Throttle Linkage
As the accelerator linkage from the pedal actuates the rod (1) it turns the primary circuit throttle plate (inside the base of the carburetor) and rotates the linkage slider (2) until it contacts the start of the secondary circuit linkage (3) which pushes the next stage of the linkage (4) to actuate the secondary throttle plate (5).

Other
This little guy operates a second throttle plate in the secondary circuit. There is a little guessing here on my part, so I'm open to corrections here again. I think the purpose of this is to smooth the transition between the circuits. It has a weighted lever and appears to be actuated solely by airflow in the carburetor throat.

Primary and Secondary Circuits (Basic)
These are two barrel progressive carburetors (as opposed to synchronous). This means there is essentially a carburetor for low RPM operation, and a entire second (albeit partial) helper carburetor for higher RPM fuel delivery needs. The operation between these two carburetors, or circuits, for this vintage carburetor is actuated via mechanical linkage (I'm given to understand some later models actuate the secondary circuit by vacuum).

Idle Circuit
The images below describe the mechanics of the idle circuits using the secondary barrel, though the same circuit exists in both barrels. The idle circuit provides fuel for lower RPM operation.
Fuel enters the idle circuit at the bottom of the bowl via the aperture pictured above courtesy of the pressure differentials described above.
It travels through the passages in the carburetor body through the slow jets. The jets are precisely sized to allow just the right amount of fuel through.
It goes up to the air horn and through the above indicated aperture into the carb throat.
The vacuum that draws fuel through the idle circuit and into the carb throat is generated by air rushing around the edges of the throttle plates past small apertures in the base of the carburetor. In the photo above, the first hole is the vacuum port for the distributor, the second one is the first that is uncovered to draw fuel into the secondary idle circuit, and the third hole back (which is somewhat difficult to see due to the lighting) is the second progression hole.
The primary circuit has a slot instead of holes for a smoother progression. The hole below the progression slot (above in the photo) is for the idle adjustment screw (additional baseline vacuum independent of throttle plate position).

Main Circuit
The main circuit is responsible for higher RPM operation. There is a main circuit in each barrel.
Fuel enters the main circuit through the main jets, which control how much fuel is allowed through.
It enters through these guys which draw the fuel up and into the carb throat. They also control the transition between the idle and the main circuit by drawing air into the metered holes in the photo above. The size of this hole plays a part in controling how soon the main circuit comes into play (bigger is sooner).
Another role of these parts is to emulsify the fuel. I believe (based on other carburetors) that the brass tubes inside the brass tubes in the photo have holes up and down them of various sizes and configurations to control the emulsification of the fuel. The level of the fuel inside the bowl in relation to these holes also plays a role in proper emulsification, and is dictated by the float adjustment.
These parts come in different sizes (different sized holes in the first photo, different sized and configured holes inside, etc.) indicated by the numbers on the tops of these (2 and 4 in the two pictured). Air rushing past the small hole (visible next to my finger in the upper left) sped up by the venturis in the same part are responsible for drawing the fuel into the carb throat.

Other BitsPower Valve
The power valve is responsible for adding additional fuel in instances of heavy loading.
In cases of high manifold vacuum (high load), vacuum is pulled through a hollow screw in the center of the carb (between the base and the body)...
...all the way to a hole in the air horn. This leads to the plunger pictured above through a passage in the air horn...
...and draws it up and off the sprung power valve (brass in the bottom center of the bowl) opening it and allowing fuel in.
It travels through passages in the carburetor to the primary main circuit and adds fuel to the mix (metered by the power jet screwed into the base of the power valve assembly).

Accelerator Pump
When you step on the throttle, more air is allowed into the cylinders. Unfortunately, air moves faster than fuel. This difference in time causes a momentary lean condition and causes the engine to stumble and accelerate slowly. To make up for this, the accelerator pump manually squirts a bit of extra fuel into the cylinders to make up for this lag.
The lever, pictured front and center, is attached directly to the accelerator linkage and is moved directly (this sorta belongs back in the linkage section, but this is hard enough to follow as is).
The lever actuates a plunger (pink above)...
That goes into this hole and draws fuel in from a hole in the bottom of the float bowl through a check valve. The top hole is an overflow relief valve necessary due to the direct mechanical linkage (accelerator pumps are often diaphragm driven), and air is allowed in from the bowl via the squared notch at the top (to make up for the movement of the piston at the top).
The fuel is pushed up through the body (past another check valve) and through the small brass nozzle at the top of the primary throat (between my thumb and the triangular gasket).

Idle Shutoff
This bad boy is the only electric part of these carbs.
It's a solenoid that, when current is applied, retracts a plunger from the passage between the progression slot at the base of the carb and the slow jet in the idle circuit. When this happens, no more fuel can enter the engine, and in combination with the ignition system losing power shuts down the engine.

Closing
That's all I got for now. There were a few things I wasn't completely sure about (as stated in the opening and at the points of doubt distributed throughout), and if I misspoke somewhere, please let me know. Additionally, if I left something out, let me know and I'll try to fill in any gaps. I'm going to submit now (I've been working on this for several hours now), and probably read through it again later when I'm fresh and probably edit a little to clean it up.

I hope this helps someone with something at some point, or at the very least was found at least marginally interesting (which must have been the case for anyone to read this far). I have a few things to do still on my 40 before it's ready to be tuned, but if there's a positive response I'll do a part 2 in the form of baseline tune, and a part 3 on fine tuning (jetting). Thanks for reading.